Year: 2026
Type: preprint
Abstract: We study the nonlinear Hamilton--Jacobi system arising as the formal semi-classical limit of the chordal Belavin--Polyakov--Zamolodchikov equations with a common eigenvalue parameter $λ$. We determine the number of global real-valued solutions modulo additive constants. When $λ=0$, the number of solutions can be derived using the connection between the solutions and rational functions, due to Eremenko. We focus on the case when $λ\neq 0$ in this article. For $λ>0$, a Schwarzian correspondence relates the algebraic equations for the gradients to quasi-exponentials with prescribed critical points. Enumeration and transversality results of Mukhin, Tarasov, and Varchenko yield smooth real branches, and a Poisson bracket identity establishes their exactness. An energy bound rules out global real-valued solutions for $λ<0$.
Source: arXiv (Cornell University)
Institution: Tsinghua University
Language: English
Topic: Nonlinear Waves and Solitons
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7215123851